
 If \(\ell \)  is slowly varying, then
[1]} \(L\)  is slowly varying with
\(\lim \limits _{r\rightarrow 0+}L(r)=+\infty \qquad \text{ and }\qquad \lim \limits _{r\rightarrow 0+}\frac{\ell (r)}{L(r)}=0\,.\) 

 If \(\ell \)  is slowly varying and \(\rho >-1\) , then
Karamata's theorem [1]} ensures
\(\lim _{r\rightarrow 0+}\frac{\int _0^rs^{\rho }\ell (s)\,ds}{r^{\rho +1}\ell (r)}=(\rho +1)^{-1}\,.\) 

 If \(\ell \)  is
regularly varying of order \(-\alpha < 0\)  (in our case \(0 < \alpha < 2\) ),
then
[1]}
\(\lim _{r\rightarrow 0+}\frac{L(r)}{\ell (r)}=\alpha ^{-1}\,.\) 
In particular, if \(\ell \)  is regularly varying of order \(-\alpha < 0\) ,
then so is \(L\) .

 Assume \(\ell \)  is regularly varying of order
\(-\alpha \le 0\)  and stays bounded away from 0 and \(+\infty \)  on every compact
subset of \((0,1)\) . Then Potter's theorem [1]} implies
that for every \(\delta >0\)  there is a
constant
\(C=C(\delta )\ge 1\)  such that for \(r,s \in (0,1)\)
\(\frac{\ell (r)}{\ell (s)}\le C\max \left\lbrace  \left(\frac{r}{s}\right)^{-\alpha -\delta },\left(\frac{r}{s}\right)^{-\alpha +\delta } \right\rbrace  \,.\) 

 Since \(L\)  is nonincreasing, we observe
\(\lim \limits _{r\rightarrow 0+} L(r) \in (0,+\infty ]\) .

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